You can hide the big play button (and show no user interface at all) by selecting none from the Controller Skin setting of the in5 Video widget (in5 > Interactive Widgets) or the Controller setting of the InDesign Media panel (Window > Interactive > Media).
It's recommended that set the video to Auto Play (or Play on Page Load within the Media panel) because the user will have no other way to start the video, unless you've created a separate button within InDesign to trigger playback of the video.
Note that several browsers have restrictions on autoplaying media that isn't trigger or preceded by user interface. To work around this, you can Mute the video in the Video widget panel and this will bypass the browser's auto play restrictions.
Make sure to use in5 version 3.8.12 or newer to take advantage of the fix for video controls in iOS that are configured to not appear.